(1) Find the coordinates of point A and point B;
(2) Let d be any point on the known parabola symmetry axis, and when the area of △ACD is equal to the area of △ACB, find the coordinates of point D;
(3) If the straight line L passes through point E (4 4,0) and M is the moving point on the straight line L, when there are only three right triangles with vertices A, B and M, the analytical expression of the straight line L is found.
Test center: Quadratic function synthesis problem.
Analysis: (1) Point A and Point B are the intersection of parabola and X axis, let y=0, and solve the quadratic equation of one variable.
(2) According to the meaning of the question, find the height of AC side in △ACD, and set it as H. In the coordinate plane, the parallel lines of AC and the distance between parallel lines are equal to H. According to the equal area of equal bottom and equal height, it can be known that the intersection of parallel lines and coordinate axis is point D. 。
From the point of view of linear function, such parallel lines can be regarded as the result of upward or downward translation of straight line AC. Therefore, the analytical expression of straight line AC can be obtained first, and then the translation distance can be obtained, thus the coordinates of point D can be obtained.
Note: There are two such parallel lines, as shown in the answer sheet 1.
(3) The key to this problem is to understand the meaning of "a right triangle with only three vertices A, B and M".
Because point A and point B are perpendicular to the X-axis, the two intersections with the straight line L can form a right triangle with point A and point B, so there are already two right triangles that meet the meaning of the question. The third right triangle is considered from the positional relationship between a straight line and a circle. When the straight line is tangent to the circle, according to the theorem of circumferential angle, the tangent point forms a right triangle with points A and B, and the problem is solved.
Note: There are two such tangents, as shown in Figure 2.
Solution: Solution: (1) Let y=0, that is, -38x2-34x+3=0.
X 1=-4，x2=2，
∴ The coordinates of point A and point B are A (-4,0) and B (2 2,0) respectively.
(2) The symmetry axis of parabola y=-38x2-34x+3 is a straight line x =-342x38 =- 1,
That is, the abscissa of point D is-1,
S△ACB= 12AB？ OC=9，
In Rt△AOC, AC=OA2+OC2=42+32=5.
Let the height of AC side in △ACD be h, then there is 12AC? H=9, and the solution is H = 185.
As shown in figure 1, a straight line in the coordinate plane is parallel to AC, and the distance to AC is =h= 185. There are two such straight lines, that is, l 1 and l2, so the two intersections of this straight line and the symmetry axis x=- 1 are the points for finding D. 。
Let l 1 intersect the Y axis at E and let c be CF⊥l 1 at F, then CF=h= 185.
∴ce=cfsin∠cef=cfsin∠oca= 18545=92.
Let the analytical formula of straight line AC be y=kx+b, and substitute the coordinates of A (-4,0) and C (0 0,3).
Get -4k+b=0b=3，get k=34b=3。
The analytical formula of linear communication is y = 34x+3.
The straight line l 1 can be regarded as the CE length unit (92 length units) and translated downward to form the straight line AC.
The analytical formula of the ∴ straight line l 1 is y = 34x+3-92 = 34x-32.
Then the ordinate of D 1 is 34×(- 1)-32=-94, ∴D 1(- 1,-94).
Similarly, if the straight line AC moves up by 92 length units to get l2, D2 (- 1, 274) can be obtained.
To sum up, the coordinates of point D are: D 1(- 1, -94) and D 2 (- 1, 274).
(3) As shown in Figure 2, let AB be the diameter ⊙F, the center of the circle is f, and there are two tangents passing through point E ⊙ f. 。
Connect FM, pass through M, and make MN⊥x axis at N point.
∫A(-4，0)，B(2，0)，
∴ f (- 1, 0), ∫f radius FM = FB = 3.
And FE=5, then in Rt△MEF,
ME=52-32=4，sin∠MFE=45，cos∠MFE=35。
In Rt△FMN, MN=MF? sin∠MFE=3×45= 125，
FN=MF？ Cos∠MFE=3×35=95, then ON=45,
∴M point coordinates are (45, 125).
The straight line l passes through m (45, 125), e (4 4,0),
Let the analytical formula of the straight line L be y=kx+b, then there is
45k+b= 1254k+b=0, and the solution is k=-34b=3.
So the analytical formula of the straight line L is y =-34x+3.
Similarly, the analytical expression of another tangent can be obtained as y = 34x-3.
To sum up, the analytical formula of the straight line L is y=-34x+3 or y = 34x-3.
Comments: The key to solve this problem is the comprehensive application of knowledge such as quadratic function, linear function and circle. The difficulty lies in the understanding of the condition that "there are only three right-angled triangles with vertices A, B and M" in problem (3), which can be solved from the positional relationship between straight lines and circles. This problem is difficult, which requires students to master what they have learned and use it flexibly.